Descriptive-complexity conjecture for nondeterministic dynamically generated basic sequences

Let Q=(qn)n1Q=(q_n)_{n\geq 1} be a nondeterministic dynamically generated basic sequence. Let N(Q)\mathcal{N}(Q), DN(Q)\mathcal{DN}(Q), and UN(Q)\mathcal{UN}(Q) denote the corresponding normality classes, and let D2(Π30)D_2(\boldsymbol{\Pi}_3^0) denote the stated difference-hierarchy completeness class. Descriptive-complexity conjecture. The following sets each have Hausdorff dimension 11 and are D2(Π30)D_2(\boldsymbol{\Pi}_3^0)-complete:

  1. (N(Q)DN(Q))UN(Q)(\mathcal{N}(Q)\cap\mathcal{DN}(Q))\setminus\mathcal{UN}(Q);
  2. DN(Q)N(Q)\mathcal{DN}(Q)\setminus\mathcal{N}(Q);
  3. N(Q)DN(Q)\mathcal{N}(Q)\setminus\mathcal{DN}(Q).

This is proposed as a strong converse direction to the deterministic normality-equivalence conjecture, predicting both maximal Hausdorff dimension and precise descriptive complexity.

Sources & referencesView supporting material

Primary source

Sohail Farhangi and Bill Mance, “The Theory of Normality for Dynamically Generated Cantor Series Expansions”, arXiv:2512.01239 (2025).

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